The main interest of this article is the one-sided boundedness local discrepancy $\alpha\in\mathbb{R}\setminus\mathbb{Q}$ on interval $(0,c)\subset(0,1)$ defined by \[D_n(\alpha,c)=\sum_{j=1}^n 1_{\{\{j\alpha\}<c\}}-cn.\] We focus special case $c\in (0,1)\cap\mathbb{Q}$. Several necessary and sufficient conditions $\alpha$ for $(D_n(\alpha,c))$ to be one-side bounded are derived. Using these, c...

BACKGROUND: Clostridium septicum has played a significant role as a causative agent of many acute fetal diseases in man and animals. Alpha- toxin is the main factor in the pathogenesis of C. septicum with hemolytic, necrotic and lethal activities. OBJECTIVES: The study was designed to evaluate alpha-toxin purification and antibody production rate against a local strain of C. septicum NH2 which ...

Let $A(G)$ and $D(G)$ be the adjacency matrix degree of $G$, respectively. For any real $\alpha \in [0,1]$, Nikiforov defined $A_{\alpha}(G)$ as \[ A_{\alpha}(G) = \alpha D(G) + (1-\alpha) A(G). \] In this paper, we generalize some previous results about $A_{1/2}$-spectral radius bicyclic graphs with a given sequence. Furthermore, characterize all extremal which have largest $A_{\alpha}$-spectr...

Let f $f$ be a noncommutative polynomial of degree m ⩾ 1 $m\geqslant 1$ over an algebraically closed field F $F$ characteristic 0. If n − $n\geqslant m-1$ and α , 2 3 $\alpha _1,\alpha _2,\alpha _3$ are nonzero elements from such that + = 0 _1+\alpha _2+\alpha _3=0$ then every trace zero × $n\times n$ matrix can written as A _1 A_1+\alpha _2A_2+\alpha _3A_3$ for some i $A_i$ in the image M ( ) ...

Let $(X, \tau)$ and $(Y, \sigma)$ be topological spaces in which no separation axioms are assumed, unless explicitly stated if $\mathcal{I}$ is an ideal on $X$.Given a multifunction $F\colon (X, \tau)\rightarrow (Y, \sigma)$, $\alpha,\beta$ operators \tau)$, $\theta,\delta$ proper $X$. We introduce study upper lower $(\alpha, \beta,\theta,\delta,\mathcal{I})$-continuous multifunctions.A said to...

Let $L_{k,\alpha}^{\mathbb{Z}}$ denote the set of all bi-infinite $\alpha$-power free words over an alphabet with $k$ letters, where $\alpha$ is a positive rational number and integer. We prove that if $\alpha\geq 5$, $k\geq 3$, $v\in L_{k,\alpha}^{\mathbb{Z}}$, $w$ finite factor $v$, then there are $\widetilde v\in L_{k,\alpha}^{\mathbb{Z}}$ letter $x$ such v$ has only finitely many occurrence...

It is proved that for any $0<\beta<\alpha$, bounded Ahlfors $\alpha$-regular space contains a $\beta$-regular compact subset embeds biLipschitzly in an ultrametric with distortion at most $O(\alpha/(\alpha-\beta))$. The bound on the asymptotically tight when $\beta\to \alpha$. main tool used proof regular form of skeleton theorem.